Mathematics

ECMM743 - Logic, Models and Sets (2018)

MODULE TITLE CREDIT VALUE Logic, Models and Sets 15 ECMM743 Dr Robert Barham (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 0 0
 Number of Students Taking Module (anticipated) 15
DESCRIPTION - summary of the module content

The goal of this module is to apply the power of mathematical reasoning to itself, and to properly examine the primary tool of Mathematics, namely logic.  This study is motivated by three things.  First, its own intrinsic interest.  Secondly, its applications to other branches of Mathematics.  We won't be able to get to any of the direct applications of logic in one term, but this course will develop your proof skills along with other indirect applications.  Finally, what logic can tell us about the nature of Mathematics.   Many philosophical questions (such as ``Could all mathematics be done by a computer?'') cannot be properly answered without understanding this material.

This module will start by introducing propositional logic and deduction in propositional logic, as well as examining some of its properties (such as completeness and soundness).  Then we will do the same for predicate calculus, and then go on to discuss some of the basics of Model Theory, showing how predicate calculus can describe mathematical structures.  We will then examine the Zermelo-Fraenkel axioms of Set Theory, and study the theory of ordinals and cardinals.

You will be expected to have taken the prerequisite ECM2711 module Groups, Rings and Fields in the second year, but a perfect recollection will not be necessary.

AIMS - intentions of the module

This module aims to give you a foundation in the study of logic, and to use that to provide an introduction to two of the branches of Logic, namely Model Theory and Set Theory.  Through this course, you will become fluent in formal logical reasoning, be able to formalise mathematical structures as first order models, and clearly and rigorously answer questions about infinity.

INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)

On successful completion of this module you should be able to:

Module Specific Skills and Knowledge

1.  perform formal proofs in both propositional logic and predicate calculus;

2.  formalise mathematical structures into first order logic;

3.  describe the properties of a first order theory;

4.  perform cardinal and ordinal transfinite arithmetic;

5.  prove statements about any of the systems mentioned in 1.-4.

Discipline Specific Skills and Knowledge

6.  state and apply definitions in the field of logic and applications;

7.  write informal proofs and translate these into formal arguments;

Personal and Key Transferable / Employment Skills and Knowledge

8.  reason logically and abstractly;

9.  communicate complicated ideas in writing, professionally and using correct mathematical notation.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Syntax of propositional logic. Deduction in propositional logic.  Properties of propositional logic.

- Syntax of predicate calculus.  Definition of model.  Deduction in predicate calculus.  Properties of predicate calculus.

- Definition of models, theories and axiomatisation.  Definition of decidability and completeness.  The Compactness Theorem.

- Introduction of ZFC.  Cardinal and Ordinal arithmetic.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 33 117 0
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled learning and teaching activities 28 Lectures Scheduled learning and teaching activities 5 Tutorials Guided independent study 12 Formative assignments Guided independent study 20 Coursework assignments Guided independent study 85 Lecture and examination preparation; wider reading

ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
Form of Assessment Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Four Exercise Sheets 3 hours each All  Written and Verbal Comments

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 20 80 0
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Exam mark. Verbal/written feedback
Coursework 10 10 hours All Mark, written comments and mark scheme.
Coursework 10 10 hours All Mark, written comments and mark scheme.

DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
Original Form of Assessment Form of Re-assessment ILOs Re-assessed Time Scale for Re-assessment
All Above Written Exam (100%) All August Ref/Def Period

RE-ASSESSMENT NOTES

Referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 50% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark

RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener

ELE: http://vle.exeter.ac.uk/

Web based and Electronic Resources:

Other Resources: